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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_1.html">
\begin{equation}
{\bf x}^{\prime}={\bf P}(t)\,{\bf x},\quad \alpha \leq t \leq \beta,\tag{6.5.1}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\({\bf P}(t)\)</span> is a variable-coefficient <span class="process-math">\(n \times n\)</span> matrix. Suppose that <span class="process-math">\({\bf x}^{(1)}(t), {\bf x}^{(2)}(t), \cdots, {\bf x}^{(n)}(t)\)</span> are <span class="process-math">\(n\)</span> linear independent solutions. (We say that they form a fundamental set solutions). Introduce</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_1.html">
\begin{equation}
{\bm \Psi}(t)=({\bf x}^{(1)}(t), {\bf x}^{(2)}(t), \cdots, {\bf x}^{(n)}(t))=\left(
\begin{array}{cccc}
{\bf x}^{(1)}_1(t) &amp;  {\bf x}^{(2)}_1(t) &amp; \cdots &amp; {\bf x}^{(n)}_1(t)\\
{\bf x}^{(1)}_2(t) &amp;  {\bf x}^{(2)}_2(t) &amp; \cdots &amp; {\bf x}^{(n)}_2(t)\\
\vdots &amp; \vdots &amp;    &amp; \vdots\\
{\bf x}^{(1)}_n(t) &amp;  {\bf x}^{(2)}_n(t) &amp; \cdots &amp; {\bf x}^{(n)}_n(t)
\end{array}
\right)\tag{6.5.2}
\end{equation}
</div>
<p class="continuation">is called the fundamental matrix for the system (<a href="" class="xref" data-knowl="./knowl/eq8_1.html" title="Equation 6.5.1">(6.5.1)</a>). The general solution of (<a href="" class="xref" data-knowl="./knowl/eq8_1.html" title="Equation 6.5.1">(6.5.1)</a>) is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf x}=C_1 {\bf x}^{(1)}+C_2 {\bf x}^{(2)}+\cdots+C_n {\bf x}^{(n)}={\bm \Psi}(t) \, {\bf c},
\end{equation*}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_1.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf c}=\left(
\begin{array}{c}
C_1\\
C_2\\
\vdots\\
C_n
\end{array}
\right).
\end{equation*}
</div>
<span class="incontext"><a href="sec6_5.html#p-278" class="internal">in-context</a></span>
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